Pacific Journal of Mathematics A TOPOLOGICAL APPROACH TO INVERSE AND REGULAR SEMIGROUPS Benjamin Steinberg Volume 208 No. 2 February 2003 PACIFIC JOURNAL OF MATHEMATICS Vol. 208, No. 2, 2003 A TOPOLOGICAL APPROACH TO INVERSE AND REGULAR SEMIGROUPS Benjamin Steinberg Work of Ehresmann and Schein shows that an inverse semi- group can be viewed as a groupoid with an order structure; this approach was generalized by Nambooripad to apply to arbitrary regular semigroups. This paper introduces the no- tion of an ordered 2-complex and shows how to represent any ordered groupoid as the fundamental groupoid of an ordered 2-complex. This approach then allows us to construct a stan- dard 2-complex for an inverse semigroup presentation. Our primary applications are to calculating the maximal subgroups of an inverse semigroup which, under our topo- logical approach, turn out to be the fundamental groups of the various connected components of the standard 2-complex. Our main results generalize results of Haatja, Margolis, and Meakin giving a graph of groups decomposition for the max- imal subgroups of certain regular semigroup amalgams. We also generalize a theorem of Hall by showing the strong em- beddability of certain regular semigroup amalgams as well as structural results of Nambooripad and Pastijn on such amal- gams. 1. Introduction. In the fifties, there were two attempts to axiomatize the underlying structure of pseudogroups of diffeomorphisms of manifolds. One approach, by Wagner (and independently by Preston [16]), was via inverse semigroups; the other, by Ehresmann, was via ordered groupoids, namely the so-called inductive groupoids popularized amongst semigroup theorists by Schein [18]. It is fas- cinating that some results are proved more easily via the inverse semigroup approach, while others are more naturally proved from the point of view of ordered groupoids. In his seminal paper [11], Nambooripad extended this equivalence to an equivalence between regular semigroups and, what we shall call in this paper, r-inductive groupoids. Recent work [15] emphasizes the importance of inverse semigroups in the theory of C∗-algebras. In this paper, building on the idea that any groupoid can be realized as the fundamental groupoid of a 2-complex, we introduce the notion of an ordered 2-complex and show that any ordered groupoid can be realized as 367 368 BENJAMIN STEINBERG the fundamental ordered groupoid of an ordered 2-complex. In the process, we introduce the notion of a presentation of an ordered groupoid. While an inverse semigroup presentation is different from an ordered groupoid presen- tation, we show how to construct an ordered groupoid presentation from any inverse semigroup presentation by constructing what we call the standard ordered 2-complex of an inverse semigroup presentation (generalizing the usual notion from group presentations). This complex is closely related to the Munn and right letter mapping representations of the inverse semigroup. The maximal subgroups of the inverse (regular) semigroup will turn out to be the fundamental groups of the connected components of an ordered 2-complex representing the semigroup. This will allow us to obtain a quick topological proof that if a maximal subgroup of a finitely presented inverse semigroup acts on the left of its R-class with finite quotient, then it is finitely presented. We also introduce the Sch¨utzenberger complex of an inverse semigroup presentation. This complex is a π1-trivial covering space of the standard ordered 2-complex of the presentation and plays a role in the theory sim- ilar to that played by the Cayley complex (which it generalizes) in group theory. The 1-skeleton of the Sch¨utzenberger complex is the union of all the Sch¨utzenberger graphs of the presentation [24]. This complex and the standard ordered 2-complex, both of which are built from semigroup the- oretic tools, provide a link between the semigroup and ordered groupoid approaches to the subject. Our main result is a structure theorem for maximal subgroups of certain amalgams. In [3], it is proved that a maximal subgroup of a full amalgam of regular semigroups has a certain graph of groups decomposition. In this paper, we are able to weaken the restrictions and, at the same time, obtain a much simpler proof. The proof of [3] relies on Bass-Serre theory whence they have to construct an action of the maximal subgroup on a tree with appropriate stabilizers; they use normal forms for the amalgamated product in order to do this. Our approach is to show that the amalgamated product can be represented by a topological space which is a segment of disconnected 2-complexes; the structure of the maximal subgroups then follows from an obvious decomposition of such a graph of 2-complexes into a graph of con- nected 2-complexes. In the process, we extend a result of Hall [4] on the strong embeddability of regular semigroup amalgams as well as structural results of Nambooripad and Pastijn [12] on such. We obtain further results for certain inverse semigroup amalgams, overlapping at times with results of Bennet [1]. We also obtain results for certain amalgams in the category of inverse semigroups and prehomomorphisms. For some related results on inverse semigroup amalgams, see [22]. A TOPOLOGICAL APPROACH TO INVERSE 369 Future work, will show that the various structure theorems for idempotent pure morphisms of inverse semigroups correspond to facts about when a morphism of ordered 2-complexes can be extended nicely to a covering. 2. Preliminaries. If A is a set, we let Ae = A ∪ A−1. Then Ae+ will denote the free semigroup and Ae∗ the free monoid with involution on A. If S is a semigroup, an element e ∈ S is an idempotent if e2 = e. The set of idempotents of S will be denoted E(S). An element s ∈ S is called regular if s = sts for some t ∈ S. A semigroup in which each element is regular is called a regular semigroup. If s and t are such that sts = s and tst = t, then s and t are said to be inverses of each other. One can show that an element is regular if and only if it has an inverse [6]. A semigroup S is called an inverse semigroup if each element s ∈ S has a unique inverse (denoted s−1) or, equivalently, if it is regular and has commuting idempotents [7]. The notation S1 is used for the monoid obtained by adding an identity to S. We refer the reader to [7] for more on inverse semigroups. In this paper, we shall primarily focus on inverse semigroups, only occasionally speaking of more general regular semigroups. We briefly recall Green’s relations [6]. If S is a semigroup, then we write 1 1 s ≤R t if sS ⊆ tS ; this is a preorder and we use R for the associated equivalence relation. The preorder ≤L and equivalence relation L are defined dually. One sets H= R ∩ L and D = R ◦ L; D is the smallest equivalence relation containing R and L. If e ∈ E(S), then the H-class of e is the maximal subgroup with identity e [6]. For an inverse semigroup, s R t if and only if ss−1 = tt−1 [7] and dually for L. If I is an inverse semigroup, there is a natural partial order on I defined by s ≤ t if s = ss−1t. This order is well-known to be compatible with the operations of multiplication and taking inverses [7]. Inverse semigroups can be viewed as a variety of unary semigroups and as such there is a free inverse semigroup on any set as well as a notion of inverse semigroup presentations; see [24] for some graph-theoretical techniques for working with inverse semigroup presentation. A morphism ϕ of partially ordered sets is called an order embedding if ϕ(x) ≤ ϕ(y) if and only if x ≤ y. A surjective order embedding is called an order isomorphism. 3. Ordered graphs. Following the Serre convention [19], we define a graph X to consist of: A set V (X) of vertices; a set E(X) of edges; an involution e 7→ e−1 on E(X); and a function d : E(X) → V (X) which selects the initial vertex of an edge (the terminology d is chosen to suggest the word domain). We define 370 BENJAMIN STEINBERG r : E(X) → V (X) by r(e) = d(e−1) (and r(e) is called the terminal vertex of e). One defines, for v ∈ V (X), Star(v) = d−1(v). In general, we shall require e−1 6= e, the exception being when we view groupoids as graphs in which case local group elements of order 2 will have this property. Our reason for choosing this unorthodox notation for graphs is because many of our graphs will, in fact, be groupoids, where the terms domain and range have obvious meanings. Graph morphisms are defined in the obvious way. If ϕ : X → Y is a graph morphism and, for each v ∈ V (X), ϕ : Star(v) → Star (ϕ(v)) is, respectively, injective, surjective, bijective, then ϕ is called, respectively, an immersion, a fibration, a covering. An ordered graph is a graph X such that: V (X) and E(X) are partially ordered sets; the involution and d preserve order (whence r preserves or- der); and if e ∈ E(X), v ∈ V (X) are such that v ≤ d(e), then there is a unique edge of X, denoted (v|e), with d(v|e) = v and (v|e) ≤ e, called the restriction of e to v. Note that this third condition implies that the edges of Star(v) are incomparable for any vertex v.